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Quadratic forms that represent almost the same primes

Author: John Voight
Journal: Math. Comp. 76 (2007), 1589-1617
MSC (2000): Primary 11E12; Secondary 11E16, 11R11
Published electronically: February 19, 2007
MathSciNet review: 2299790
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Abstract | References | Similar Articles | Additional Information

Abstract: Jagy and Kaplansky exhibited a table of $ 68$ pairs of positive definite binary quadratic forms that represent the same odd primes and conjectured that their list is complete outside of ``trivial'' pairs. In this article, we confirm their conjecture, and in fact find all pairs of such forms that represent the same primes outside of a finite set.

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  • [BS] Z.I. Borevich and I.R. Shafarevich, Number theory, Academic Press, New York, 1966. MR 0195803 (33:4001)
  • [Ch] S. Chowla, An extension of Heilbronn's class number theorem, Quart. J. Math. Oxford Ser. 5 (1934), 304-307.
  • [Cox] David A. Cox, Primes of the form $ x^2+ny^2$: Fermat, class field theory, and complex multiplication, John Wiley & Sons, Inc., New York, 1989. MR 1028322 (90m:11016)
  • [D] Harold Davenport, Multiplicative number theory, third ed., Graduate texts in mathematics, vol. 74, Springer-Verlag, New York, 2000. MR 1790423 (2001f:11001)
  • [J] Gerald Janusz, Algebraic number fields, second ed., Graduate studies in mathematics, vol. 7, American Mathematical Society, Providence, RI, 1996. MR 1362545 (96j:11137)
  • [JK] William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes, preprint.
  • [La] Serge Lang, Algebraic number theory, 2nd ed., Graduate studies in mathematics, vol. 110, Berlin: Springer-Verlag, 1994. MR 1282723 (95f:11085)
  • [Lo] Stéphane Louboutin, Minorations (sous l'hypothèse de Riemann généralisée) des nombres de classes de corps quadratiques imaginaires. Application, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 795-800. MR 1058499 (91e:11126)
  • [N] Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Springer-Verlag, Berlin, 1999. MR 1697859 (2000m:11104)
  • [S] C.L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 83-86.
  • [T] Tikao Tatuzawa, On a theorem of Siegel, Japan. J. Math. 21 (1951), 163-178. MR 0051262 (14:452c)
  • [Wa] Lawrence C. Washington, Introduction to cyclotomic fields, second ed., Graduate texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575 (97h:11130)
  • [We] P.J. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117-124. MR 0313221 (47:1776)

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Additional Information

John Voight
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
Address at time of publication: Institute for Mathematics and its Applications, 400 Lind Hall, 237 Church Street, University of Minnesota, Minneapolis, Minnesota 55455

Keywords: Binary quadratic forms, number theory
Received by editor(s): September 16, 2005
Received by editor(s) in revised form: July 25, 2006
Published electronically: February 19, 2007
Additional Notes: The author’s research was partially supported by an NSF Graduate Fellowship. The author would like to thank Hendrik Lenstra, Peter Stevenhagen, and the reviewer for their helpful comments, as well as William Stein and the MECCAH cluster for computer time
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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